Question

If $\dfrac{4}{5}$, a, 2 are three consecutive terms of an A.P., then find the value of a?

Answer 1

Hint: As the given sequence is in A.P., so we will use the definition of arithmetic series to solve this question. We will calculate the common difference between the terms and form the equations. Then by simplifying the obtained equations we get the desired answer.

Complete step by step solution:
We have been given that $\dfrac{4}{5}$, a, 2 are three consecutive terms of an A.P.
We have to find the value of a.
Now, let us assume that the ${{a}_{1}}=\dfrac{4}{5}$, ${{a}_{2}}=a$ and ${{a}_{3}}=2$.
Now, we know that the common difference between the terms of an A.P. remains the same for the whole series. Also for three consecutive terms to be in A.P. the common difference between first two terms is equal to the difference between last two terms.
Now, calculating the common difference between the terms we will get
\[\Rightarrow {{a}_{2}}-{{a}_{1}}={{a}_{3}}-{{a}_{2}}\]
Now, substituting the values we will get
$\Rightarrow a-\dfrac{4}{5}=2-a$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow a+a=2+\dfrac{4}{5} \\
 & \Rightarrow 2a=\dfrac{10+4}{5} \\
 & \Rightarrow 2a=\dfrac{14}{5} \\
 & \Rightarrow a=\dfrac{14}{5\times 2} \\
 & \Rightarrow a=\dfrac{7}{5} \\
\end{align}$

Hence we get the value of a as $\dfrac{7}{5}$.

Note: The point to be noted is that we have to check the common difference for all the terms. We can also verify the answer by using the formula of arithmetic sequence. We have three types of sequences i.e. arithmetic sequence, geometric sequence and harmonic sequence. So before solving the questions please check which kind of sequence is given in the question.